Main Question or Discussion Point

1. I read these articles, but how it is possible to say more clearly that gravitational energy does not exist?
2. What this means on an example of a Schwarchild black hole? It is a simpler example than other generalized derivation in the first link.
3. What this means on an example of a dense spherical mass,which is not black hole and it is not rotating?
4. Does exist still simpler example for visualisation and clarification of the above claims?

Einstein made the gigantic step towards the profound understanding of grav-
ity as spacetime curvature but even he was unable to accept all implications of
the revolutionary view of gravitational phenomena. It was he who first tried to
insert the concepts of gravitational energy and momentum forcefully into general
relativity in order to ensure that gravity can still be regarded as some interaction
despite that the mathematical formalism of general relativity itself refused to yield
a proper (tensorial) expression for gravitational energy and momentum. This refusal is fully consistent with the status of gravity as non-Euclidean spacetime
geometry (not a force) in general relativity. The non-existence of gravitational
force implies the non-existence of gravitational energy as well since gravitational
energy presupposes gravitational force (gravitational energy = work due to gravity = gravitational force times distance).

He connects non-tensorial nature with no existence of gravitational energy?

Einstein made the gigantic step towards the profound understanding of grav-
ity as spacetime curvature but even he was unable to accept all implications of
the revolutionary view of gravitational phenomena. It was he who first tried to
insert the concepts of gravitational energy and momentum forcefully into general
relativity in order to ensure that gravity can still be regarded as some interaction
despite that the mathematical formalism of general relativity itself refused to yield
a proper (tensorial) expression for gravitational energy and momentum. This refusal is fully consistent with the status of gravity as non-Euclidean spacetime
geometry (not a force) in general relativity. The non-existence of gravitational
force implies the non-existence of gravitational energy as well since gravitational
energy presupposes gravitational force (gravitational energy = work due to gravity = gravitational force times distance).

He connects non-tensorial nature with no existence of gravitational energy?

You should be aware that the views on GR in this essay are disputed by virtually all GR experts. The consensus is:

- The geodesic hypothesis is not really part of GR. Einstein 'retracted' as a separate assumption by virtue of his work with Infeld and Hoffman showing that it is a consequence (to a very good approximation) of the field equations themselves. Today, this is universally accepted - geodesic motion is a consequence of the field equations, not a separate assumption.

- The idea that mutually orbiting point masses follow geodesics cannot be formulated mathematically - geodesics of what geometry?

- Gravitational radiation as consequence of binary orbit has now been derived so many ways, it is essentially 'fringe' to dispute it as a consequence of GR (disputing it because you don't think it exists, or favor some other explanation of Taylor-Hulse pulsar is anothere matter).

A review of deriving motion from the field equations alone, including deviations from exact geodesic motion:

http://relativity.livingreviews.org/Articles/lrr-2011-7/ [Broken]

Given that the above derives motion from the field equations, adding an independent geodesic hypothesis produces a mathematically inconsistent theory: EFE and the geodesic hypothesis make conflicting predictions.

1. I read these articles, but how it is possible to say more clearly that gravitational energy does not exist?
2. What this means on an example of a Schwarchild black hole? It is a simpler example than other generalized derivation in the first link.
3. What this means on an example of a dense spherical mass,which is not black hole and it is not rotating?
4. Does exist still simpler example for visualisation and clarification of the above claims?

"Existence" is a rather philosophical question. The topic of energy conservation in GR is a rather highly technical one.

It's certainly true that it "doesn't exist" in the sense that we're used to from classical field theory. I could add that t his is the sense in which the energy density transforms as a tensor density, but I'm not sure how helpful that would be.

However, there are important special cases, like the static geomtery of black holes, where we DO have good notions of energy.

On the other hand, there are some important cases (like cosmology) where trying to apply familar concepts of energy conservation will get you into deep trouble - like asking where the energy goes when radiation gets red-shifted.